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Full text of "Treatise On Analysis Vol-Ii"

9   EXAMPLES OF CONVOLUTIONS OF MEASURES AND FUNCTIONS       285

PROBLEMS

1.    Let G be a locally compact group and p, a positive nonzero bounded measure on G,
such that jit * JJL = p..

(a)    Show that S = Supp(^) is compact. (If /e 3f+(G) is not identically zero, remark
that fji */must be constant on S, and use (14.9.2(iii)).)

(b)    Show that S is a compact subgroup of G and that p, is the Haar measure on S for
which /x(S) = 1. (Use (a), Problem 2 of Section 12.9 and Problem 5 of Section 14.7.)

2.    (a)   Generalize (14.9.2(i)) to the case where 1 < p < + oo, by using Holder's inequal-
ity (Section 13.11, Problem 12).

(b)    Let ju- be a bounded measure on G. Show that the norm of the continuous endo-
morphism of LX(G, ]8) induced by /H-/A */ (14.9.2) is equal to \\fi\\. (Let (/) be a
sequence satisfying the conditions of (14.11.1). If the norm in question were \\p\\ -a
with a > 0, we should have N^/x, */ * g)  (|W - oWdfn * ff) ^ (HHI - a)Nito).
Deduce that Nao(ju, */) < ||jLt|| a, and obtain a contradiction by letting n tend
to +00.)

(c)    Under the hypotheses of (b), show that the norm of the continuous endomorphism
of L(G, j3) induced by /*-*/* */ is equal to \\p.\\. (Reduce to the case where p. has
compact support and has a continuous density with respect to |JK,|.)

(d)    Suppose that G is compact and JJL is positive. Show that, for 1 <p < -f oo, the
norm of the continuous endomorphism of Lp(G,/2) induced by f\/A*/ is equal

to HI.

(e)    Let G be a cyclic group of order 3. Give an example of a measure p, on G such
that the norm of the endomorphism of L2(G, p) induced by f\-+p */is strictly less
than ||/* ||.

3.    (a)   Let p, be a bounded measure on G. Show that we have NI(JLI */) = NA(/) for all
/ e JS?l(G, ]3) if and only if //, is of the form c - ex with  c| = 1. (Using Problem 2(b),

show that for each /e jf(G) we must have I f fdp = f \f\d\fji\. Deduce first that

IJ                   J

^ = c|fi.|, where c is a constant such that |c|  1, and then that yu is a point-measure.)
(b) Take G to be a cyclic group of order 3. Give an example of a measure JJL on G
which is not a point-measure and is such that N2(/x */) = N2(/) for all real-valued
functions/on G.

4.    Let G be a locally compact group and ft a left Haar measure on G. Let/be a bounded
real-valued function on G, uniformly continuous with respect to a left-invariant
distance on G. If /LA is any bounded measure on G, show that /x */(relative to ]8) is
uniformly continuous with respect to a left-invariant distance on G.

5.    Let G be a locally compact group and ]S a left Haar measure on G.

(a)    With the notation of Section 13.20, Problem 1, let (p,n) be a sequence of bounded
measures on G which converges to /x with respect to the topology ^"3, and let (/)
be a sequence of functions in ^>1(G, j8) such that the sequence of bounded measures
(/  j8) converges to /  ]8 with respect to the topology $~6. Show that the sequence
of bounded measures (/An * (/  j8)) converges to (/u,B * (/  j8)) with respect to ^6
(cf. Problem 4(d) of Section 14.7). (Use Problems 1 and 2 of Section 13.20.)

(b)    Let E5/2 denote the space of bounded real-valued functions on G which are
uniformly continuous with respect to a left-invariant distance on G, and let ^5/2